Let $n$ be a positive integer. How can I show that the relation $R$ on the set of all polynomials with real–valued coefficients consisting of all pairs $(f, g)$ such that $f^{(n)}(x) = g^{(n)}(x)$ is an equivalence relation. Note that $f^{(n)}$ and $g^{(n)}$ are the $n$-th derivative of $f(x)$.

1 Answer
1

There is a general principle, that says that any relation $R$ expressed in terms of $f R g$ if and only if $f$ and $g$ have the same (whatever) is an equivalence relation. Just spell aloud the three properties, and you'll see what I mean

More formally, if $A, C$ are sets, with $A \ne \emptyset$, and $\varphi: A \to C$ is a map, then the relation $R$ on $A$ defined, for $a, b \in A$, by
$$
\text{$a R b$ if and only if $\varphi(a) = \varphi(b)$}
$$
is an equivalence relation.